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A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n. 2
0, 1, 3, 6, 17, 43, 147, 458, 1729, 6445, 27011 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

The a(1) = 1 through a(4) = 17 multiset partitions:

  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}

         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}

         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}

                    {{1},{1,1}}    {{1,1,2,3}}

                    {{1},{1,2}}    {{1,2,3,4}}

                    {{1},{1},{1}}  {{1},{1,1,1}}

                                   {{1,1},{1,1}}

                                   {{1},{1,1,2}}

                                   {{1,1},{1,2}}

                                   {{1},{1,2,2}}

                                   {{1},{1,2,3}}

                                   {{1,2},{1,3}}

                                   {{2},{1,1,2}}

                                   {{1},{1},{1,1}}

                                   {{1},{1},{1,2}}

                                   {{1},{2},{1,2}}

                                   {{1},{1},{1},{1}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];

strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];

Table[Sum[Length[Select[mps[m], And[mensity[#]==-1, Length[csm[#]]==1]&]], {m, strnorm[n]}], {n, 0, 8}]

CROSSREFS

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.

Sequence in context: A143363 A216878 A237670 * A006081 A099511 A204517

Adjacent sequences:  A321224 A321225 A321226 * A321228 A321229 A321230

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 31 2018

STATUS

approved

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Last modified January 21 01:45 EST 2022. Contains 350473 sequences. (Running on oeis4.)